## Publications

10 results found

Feyzbakhsh S, Li C, 2021, Higher rank Clifford indices of curves on a K3 surface, *SELECTA MATHEMATICA-NEW SERIES*, Vol: 27, ISSN: 1022-1824

Feyzbakhsh S, 2020, Mukai’s program (reconstructing a K3 surface from a curve) via wall-crossing, *Journal für die reine und angewandte Mathematik*, Vol: 2020, Pages: 101-137, ISSN: 0075-4102

Let C be a curve of genus g=11 or g≥13 on a K3 surface whose Picard group is generated by the curve class [C]. We use wall-crossing with respect to Bridgeland stability conditions to generalise Mukai’s program to this situation: we show how to reconstruct the K3 surface containing the curve C as a Fourier–Mukai transform of a Brill–Noether locus of vector bundles on C.

Feyzbakhsh S, Thomas RP, 2019, An application of wall-crossing to Noether-Lefschetz loci, Publisher: arXiv

Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Giesekerconjecture of Bayer-Macr\`{i}-Toda (such as $\mathbb P^3$, the quinticthreefold or an abelian threefold). Let $L$ be a line bundle supported on a very positive surface in $X$. If$c_1(L)$ is a primitive cohomology class then we show it has very negativesquare.

Bayer A, Beentjes S, Feyzbakhsh S, et al., The desingularization of the theta divisor of a cubic threefold as a moduli space

We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaveson a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$is smooth and of dimension four. Moreover, the Abel-Jacobi map to theintermediate Jacobian of $X$ maps it birationally onto the theta divisor$\Theta$, contracting only a copy of $X \subset \overline{M}_X(v)$ to thesingular point $0 \in \Theta$. We use this result to give a new proof of a categorical version of theTorelli theorem for cubic threefolds, which says that $X$ can be recovered fromits Kuznetsov component $\operatorname{Ku}(X) \subset\mathrm{D}^{\mathrm{b}}(X)$. Similarly, this leads to a new proof of thedescription of the singularity of the theta divisor, and thus of the classicalTorelli theorem for cubic threefolds, i.e., that $X$ can be recovered from itsintermediate Jacobian.

Feyzbakhsh S, Thomas RP, Rank $r$ DT theory from rank $0$

Fix a Calabi-Yau 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture ofBayer-Macr\`i-Toda, such as the quintic 3-fold. We express Joyce's generalisedDT invariants counting Gieseker semistable sheaves of any rank $r\ge1$ on $X$in terms of those counting sheaves of rank 0 and pure dimension 2. The basic technique is to reduce the ranks of sheaves by replacing them bythe cokernels of their Mochizuki/Joyce-Song pairs and then use wall crossing tohandle their stability.

Feyzbakhsh S, Thomas RP, Rank $r$ DT theory from rank $1$

Fix a Calabi-Yau 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture ofBayer-Macr\`i-Toda, such as the quintic 3-fold. We express Joyce's generalised DT invariants counting Gieseker semistablesheaves of any rank $r$ on $X$ in terms of those counting sheaves of rank 1. Bythe MNOP conjecture they are therefore determined by the Gromov-Witteninvariants of $X$.

Feyzbakhsh S, Thomas RP, Curve counting and S-duality

We work on a projective threefold $X$ which satisfies the Bogomolov-Giesekerconjecture of Bayer-Macr\`i-Toda, such as $\mathbb P^3$ or the quinticthreefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ aresmooth bundles over Hilbert schemes of ideal sheaves of curves and points in$X$. When $X$ is Calabi-Yau this gives a simple wall crossing formula expressingcurve counts (and so ultimately Gromov-Witten invariants) in terms of counts ofD4-D2-D0 branes. These latter invariants are predicted to have modularproperties which we discuss from the point of view of S-duality andNoether-Lefschetz theory.

Feyzbakhsh S, Pertusi L, Serre-invariant stability conditions and Ulrich bundles on cubic threefolds

We prove a general criterion which ensures that a fractional Calabi--Yaucategory of dimension $\leq 2$ admits a unique Serre-invariant stabilitycondition, up to the action of the universal cover of$\text{GL}^+_2(\mathbb{R})$. We apply this result to the Kuznetsov component$\text{Ku}(X)$ of a cubic threefold $X$. In particular, we show that all theknown stability conditions on $\text{Ku}(X)$ are invariant with respect to theaction of the Serre functor and thus lie in the same orbit with respect to theaction of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As anapplication, we show that the moduli space of stable Ulrich bundles of rank$\geq 2$ on $X$ is irreducible, answering a question asked by Lahoz, Macr\`iand Stellari.

Feyzbakhsh S, An effective restriction theorem via wall-crossing and Mercat's conjecture

We prove an effective restriction theorem for stable vector bundles $E$ on asmooth projective variety: $E|_D$ is (semi)stable for all irreducible divisors$D \in |kH|$ for all $k$ greater than an explicit constant. As an application, we show that Mercat's conjecture in any rank greater than$2$ fails for curves lying on K3 surfaces. Our technique is to use wall-crossing with respect to (weak) Bridgelandstability conditions which we also use to reprove Camere's result on slopestability of the tangent bundle of $\mathbb{P}^n$ restricted to a K3 surface.

Feyzbakhsh S, Mukai's Program (reconstructing a K3 surface from a curve) via wall-crossing, II

Let $C$ be a curve on a K3 surface $X$ with Picard group $\mathbb{Z}.[C]$.Mukai's program seeks to recover $X$ from $C$ by exhibiting it as aFourier-Mukai partner to a Brill-Noether locus of vector bundles on $C$. We usewall-crossing in the space of Bridgeland stability conditions to prove this forgenus $\ge14$. This paper deals with the case $g-1$ prime left over from PaperI.

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